Geometry of multilinear forms on Rm with a certain norm

초록

For every m >= 2, let R-parallel to.parallel to(m) be R-m with a norm parallel to.parallel to( )such that vertical bar ext B-Rm(parallel to.parallel to) vertical bar= 2m. For every n >= 2, we devote ourselves to the description of the sets of extreme and exposed points of the closed unit balls of L(R-n(parallel to.parallel to)m) and L-s(R-n(parallel to.parallel to)m), where L(R-n(parallel to.parallel to)m) is the space of n-linear forms on R-parallel to.parallel to(m), and L-s(R-n(parallel to.parallel to)m) is the subspace of L(R-n(parallel to.parallel to)m) consisting of symmetric n-linear forms. Let F = L(R-n(parallel to.parallel to)m) or L-s(R-n(parallel to.parallel to)m). First we classify the extreme and exposed points of the closed unit ball of F. We obtain vertical bar ext B-L(nR parallel to.parallel to m()) vertical bar = 2((mn)) and vertical bar ext B-Ls(nR parallel to.parallel to m) vertical bar = 2( )(dim(Ls(nR parallel to)(vertical bar.parallel to vertical bar)(m))). We also show that every extreme point of the closed unit ball of F is exposed. It is shown that ext B-Ls ((nR parallel to.parallel to m)) = ext B-L((nR parallel to.parallel to m)) boolean AND L-s (R-n(parallel to.parallel to)m) and exp B-Ls(nR parallel to.parallel to m) = exp B-L((nR parallel to.parallel to m)) boolean AND L-s(R-n(parallel to.parallel to)m).

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